Analog electronic devices all have a nonlinear behavior to some extent. In particular, this non-linearity can reduce the bandwidth of certain instruments when large signals are applied, if the absolute value of the rate of change of the signals approaches the slew rate limit of the electronic components constituting the amplifier. This can result in an inaccurate representation of the signal present at the input of the analog electronic devices, such as (but not limited to) the front end amplifiers of digital storage oscilloscopes.
While look-up tables can be used to correct non-linearities present at low frequency and in the direct current (DC) case, compression, being a high frequency phenomenon, cannot be corrected by simple look-up tables, as recognized by the inventors of the present invention.
It is well known that Volterra equations can be employed to describe many dynamic non-linear phenomena by posing a non-linear differential equation that the high frequency signal follows, but in practice, in addition to the difficulty of the choice of the relevant equations, the solution is excessively complex for a real-time computation. Moreover, in the case of a digital storage oscilloscope (DSO) where the signal may not be repetitive, the presence or absence of signals beyond the Nyquist frequency cannot be ascertained. Furthermore, for the correction of non-linearities in large high frequency signals, the solution of Volterra equations requires the knowledge of the harmonics of this signal. Aliasing removes this knowledge. The method of “Volterra kernels” is distinct from a full-fledged solution of the Volterra Ordinary Differential Equations. Volterra kernels that involve a time difference td larger than or equal to the sampling period τ can actually be used on regularly-spaced samples, and provides a means to compensate for the reduction of bandwidth at large signal amplitude, by summing a power series of non-linear terms that are related to differences between samples. One could then, in principle, establish the values a3 such that the non-linearity correction Σj=1Naj(f(t)−f(t−td)))j summed to a signal f(t) reproduces the data. However, the inventors of the present invention have recognized that this well-known method has an intrinsic problem with the description of the “limit” or asymptotic behavior of the signal. Because this method uses a finite order polynomial N, it cannot accurately describe a non-linearity that nears a finite limit as the stimulus approaches plus or minus infinity, neither can it accurately describe a non-linearity that diverges to plus or minus infinity when the stimulus approaches a specific value. This is a serious shortcoming of this method because errors increase considerably in the description of the circuits behavior as one approaches the slew rate limit.
A related problem is that such fit procedures have too many degrees of freedom, in fact as many degrees of freedom as the order of the polynomial used; unless extensive data taking takes place for each setting of the amplifier, the fit is under-constrained, and statistical and systematic errors develop. This is especially an acute problem in the context of a DSO front-end amplifier, which has very many possible settings (e.g. over 300 gain settings in the variable gain stage). Yet another method, the method of look-up tables, represents an even larger level of complexity. As an example, a valid compression correction method for an 8-bit analog-to-digital converter (ADC) system would be a correction based on pairs of consecutive samples. It would require 256×256=65536 parameters, that is 65536 parameters for each range (variable gain) of each channel. As recognized by the inventors of the present invention, such a method is clearly too complex for a practical implementation.
Another method for correcting non-linearities in a received signal is to introduce several quadratic terms representing the products of the current sample with the nth preceding sample as suggested by U.S. Utility Pat. No. 6,687,235 B1, filed Jan. 21, 2000, titled “Mitigation of non-linear signal perturbations using truncated Volterra-based Non-Linear Echo Canceler” to Chu. The Chu method is interesting as it uses consecutive sampling values to calculate a correction in a causal process. However, quadratic coefficients are not sufficient to handle the non-linearities, only cubic terms (and other “odd” non-linearities) can account for an identical absolute value of the slew rate limit in positive and negative going transitions, which is a feature of most DSO front-end amplifiers. Also, this method does not provide exact linearity for DC signals, which is a problem because most DSO front-end amplifiers have better than 1.5% Integral Non-linearity.
Another approach is to consider, as in US Patent Application US2009/0058521 A1, filed Aug. 31, 2007 titled “System and Method of Digital Linearization In Electronic Devices” to Fernandez, a correction of the non-linearity which is a multivariable polynomial function of selected variables such as the value of the measured signal at a specific time, the value of the measured signal at a different time, and the local derivative of the signal. Once again, the inventors of the present invention have determined that for the same reason that finite-order Volterra kernels fail at reproducing accurately asymptotic behavior, this method also does not work accurately in the case of large stimuli that are near the slew rate limit. Likewise, even though a judicious choice of polynomial coefficients can yield approximate linearity for DC signals, it does not provide exact linearity for DC signals.
Several existing methods provide a correction for the amplitude of compressed alternating current (AC) signals, but do not address the correction of the phase. It is obvious to a person skilled in the art that a slew rate limitation results in the signal lagging in phase (occurring later) with respect to a signal not affected by the slew rate limitation. For some applications, decompression, in other words non-linear enhancement of certain frequencies and not others, is a process in which an exact preservation of phase is not important, as in U.S. Utility Pat. No. 5,349,389, titled “Video attenuator with output combined with signal from non-linear shunt branch to provide gamma correction and high frequency detail enhancement”, filed Apr. 14, 1993, to Keller. Some others have given importance to the question of phase shift when non-linearities are present. U.S. Utility Pat. No. 6,344,810, titled “Linearity Error compensator”, filed Jan. 18, 2001, to Velazquez, teaches a way to emulate, and correct for, a phase-shifted distortion signal by applying distinct filtering to each power of a polynomial series of the signal—the fundamental, the square of the signal, the cube of the signal, and so on. As recognized by the inventors of the present invention, this method creates a problem because the phase shifting is implemented in specialized units that create a distinct frequency-dependent phase shift for each power of the signal in the correction. In this method, a non-zero correction at high frequency will necessarily impact the linearity for DC signals—in contradiction with the excellent linearity present in DSO front-end amplifiers. Also, like the other methods presented here, the method cannot reproduce accurately the asymptotic behavior of slew rate limit, because it fits non-linearity with a polynomial of finite order.
In U.S. Utility Pat. No. 6,911,925 B1, titled “Linearity Compensation by harmonic Cancellation”, filed Apr. 2, 2004 to Slavin, an enhancement to the above invention by Velazquez is suggested; like the other methods described above, it lacks the capability to reconcile the linearity of DC signals with the wanted AC signals correction. Since it uses a finite set of powers of the signal for correction, it also lack the capability to accurately correct for non-linearity when the signal approaches an asymptotic limit, like the other methods shown above. It includes a stage of amplitude and phase correction prior to the stages of non-linear linearity correction, which amounts to a prefiltering. This prefiltering is useful to reduce the amount of tones, arising from harmonics, which may end up in band due to aliasing. However, any such prefiltering reduces the usable bandwidth. This problem is more acute for analog stages having a bandwidth close to the Nyquist frequency of the sampling system. This is part of a general problem recognized by the inventors of the present invention as being shared by all the methods so far: the correction of the harmonics is possible but in general some of the harmonics can alias back into the bandwidth of the amplifier where no harmonic to be offset is present due to the roll-off of the amplifier. The very process of correcting for the non-linearity can create new non-linearities, and if these alias back into the band of the amplifier, and do not exactly cancel a preexisting artifact, no filtering can remove it.
Another side effect of the above method of harmonic cancellation is the change in the frequency response of the signal originating from the amplitude and phase correction filter, as well as the change in the magnitude of the fundamental originating for terms having the fundamental frequency being present in the power expansion. Both these effects change the Bode Plot with respect to the Bode Plot that would be obtained without the cancellation processing and in the small signal limit. This is a serious inconvenience for DSO channels where a precisely prescribed frequency response for a sweep of sine waves is desired.
In summary, as recognized by the inventors of the present invention, while a number of methods exist in the prior art for the correction of AC non-linearities, many of these methods use analog electronics before the digitization, and those which address digital signal processing (DSP) methods use look up tables which cannot be filled accurately within a reasonable calibration time, and fits to polynomials which cannot correctly handle the asymptotic response. With few exceptions, these prior art methods do not address the phase shifts that must be compensated to faithfully represent the original signals, and those that do use algorithms such that the reproduction of non-linearities at high frequency creates discrepancies with the linearity for DC signals. These methods also can create excessive in-band aliased harmonics unless the input signal bandwidth is severely limited. All three of these phenomena render these methods useless for the faithful decompression of signals coming out of a high bandwidth front-end amplifier such as those present in DSO channels.